theorem Th19:
  I/\/FI is 1_Lattice & Top (I/\/FI) = (Top I)/\/FI
proof
  set L = I/\/FI;
  set R = equivalence_wrt FI;
  set x = (Top I)/\/FI;
A1: now
    let y be Element of L;
    L = LattStr (#Class R, (the L_join of I)/\/R, (the L_meet of I)/\/R #)
    by Def5;
    then consider j such that
A2: y = Class(R,j) by EQREL_1:36;
A3: (Top I)"\/"j = Top I;
A4: y = j/\/FI by A2,Def6;
    hence x"\/"y = x by A3,Th15;
    thus y"\/"x = x by A3,A4,Th15;
  end;
  hence I/\/FI is 1_Lattice by LATTICES:def 14;
  hence thesis by A1,LATTICES:def 17;
end;
